This book presents some facts and methods of the Mathematical Control Theory treated from the geometric point of view. The book is mainly based on graduate courses given by the first coauthor in the years 2000-2001 at the International School for Advanced Studies, Trieste, Italy. Mathematical prerequisites are reduced to standard courses of Analysis and Linear Algebra plus some basic Real and Functional Analysis. No preliminary knowledge of Control Theory or Differential Geometry is required. What this book is about? The classical deterministic physical world is described by smooth dynamical systems: the future in such a system is com­ pletely determined by the initial conditions. Moreover, the near future changes smoothly with the initial data. If we leave room for "free will" in this fatalistic world, then we come to control systems. We do so by allowing certain param­ eters of the dynamical system to change freely at every instant of time. That is what we routinely do in real life with our body, car, cooker, as well as with aircraft, technological processes etc. We try to control all these dynamical systems! Smooth dynamical systems are governed by differential equations. In this book we deal only with finite dimensional systems: they are governed by ordi­ nary differential equations on finite dimensional smooth manifolds. A control system for us is thus a family of ordinary differential equations. The family is parametrized by control parameters.

1 Vector Fields and Control Systems on Smooth Manifolds 1 1.1 Smooth Manifolds 1 1.2 Vector Fields on Smooth Manifolds 4 1.3 Smooth Differential Equations and Flows on Manifolds 8 1.4 Control Systems 12 2 Elements of Chronological Calculus 21 2.1 Points, Diffeomorphisms, and Vector Fields 21 2.2 Seminorms and $C^{\infty }(M)$-Topology 25 2.3 Families of Functionals and Operators 26 2.4 Chronological Exponential 28 2.5 Action of Diffeomorphisms on Vector Fields 37 2.6 Commutation of Flows 40 2.7 Variations Formula 41 2.8 Derivative of Flow with Respect to Parameter 43 3 Linear Systems 47 3.1 Cauchy's Formula for Linear Systems 47 3.2 Controllability of Linear Systems 49 4 State Linearizability of Nonlinear Systems 53 4.1 Local Linearizability 53 4.2 Global Linearizability 57 5 The Orbit Theorem and its Applications 63 5.1 Formulation of the Orbit Theorem 63 5.2 Immersed Submanifolds 64 5.3 Corollaries of the Orbit Theorem 66 5.4 Proof of the Orbit Theorem 67 5.5 Analytic Case 72 5.6 Frobenius Theorem 74 5.7 State Equivalence of Control Systems 76 6 Rotations of the Rigid Body 81 6.1 State Space 81 6.2 Euler Equations 84 6.3 Phase Portrait 88 6.4 Controlled Rigid Body: Orbits 90 7 Control of Configurations 97 7.1 Model 97 7.2 Two Free Points 100 7.3 Three Free Points 101 7.4 Broken Line 104 8 Attainable Sets 109 8.1 Attainable Sets of Full-Rank Systems 109 8.2 Compatible Vector Fields and Relaxations 113 8.3 Poisson Stability 116 8.4 Controlled Rigid Body: Attainable Sets 118 9 Feedback and State Equivalence of Control Systems 121 9.1 Feedback Equivalence 121 9.2 Linear Systems 123 9.3 State-Feedback Linearizability 131 10 Optimal Control Problem 137 10.1 Problem Statement 137 10.2 Reduction to Study of Attainable Sets 138 10.3 Compactness of Attainable Sets 140 10.4 Time-Optimal Problem 143 10.5 Relaxations 143 11 Elements of Exterior Calculus and Symplectic Geometry 145 11.1 Differential 1-Forms 145 11.2 Differential $k$-Forms 147 11.3 Exterior Differential 151 11.4 Lie Derivative of Differential Forms 153 11.5 Elements of Symplectic Geometry 157 12 Pontryagin Maximum Principle 167 12.1 Geometric Statement of PMP and Discussion 167 12.2 Proof of PMP 172 12.3 Geometric Statement of PMP for Free Time 177 12.4 PMP for Optimal Control Problems 179 12.5 PMP with General Boundary Conditions 182 13 Examples of Optimal Control Problems 191 13.1 The Fastest Stop of a Train at a Station 191 13.2 Control of a Linear Oscillator 194 13.3 The Cheapest Stop of a Train 197 13.4 Control of a Linear Oscillator with Cost 199 13.5 Dubins Car 200 14 Hamiltonian Systems with Convex Hamiltonians 207 15 Linear Time-Optimal Problem 211 15.1 Problem Statement 211 15.2 Geometry of Polytopes 212 15.3 Bang-Bang Theorem 213 15.4 Uniqueness of Optimal Controls and Extremals 215 15.5 Switchings of Optimal Control 218 16 Linear-Quadratic Problem 223 16.1 Problem Statement 223 16.2 Existence of Optimal Control 224 16.3 Extremals 227 16.4 Conjugate Points 229 17 Sufficient Optimality Conditions, Hamilton-Jacobi Equation,Dynamic Programming 235 17.1 Sufficient Optimality Conditions 235 17.2 Hamilton-Jacobi Equation 242 17.3 Dynamic Programming 244

Andrei A. Agrachev

Born in Moscow, Russia.

Graduated: Moscow State Univ., Applied Math. Dept., 1974.

Ph.D.: Moscow State Univ., 1977.

Doctor of Sciences (habilitation): Steklov Inst. for Mathematics, Moscow, 1989.

Invited speaker at the International Congress of Mathematicians ICM-94 in Zurich.

Over 90 research papers on Control Theory, Optimization, Geometry (featured review of Amer. Math. Soc., 2002).

Professional Activity: Inst. for Scientific Information, Russian Academy of Sciences, Moscow, 1977-1992; Moscow State Univ., 1989-1997; Steklov Inst. for Mathematics, Moscow, 1992-present; International School for Advanced Studies (SISSA-ISAS), Trieste, 2000-present.

Current positions: Professor of SISSA-ISAS, Trieste, Italy

and Leading Researcher of the Steklov Ins. for Math., Moscow, Russia

Yuri L. Sachkov

Born in Dniepropetrovsk, Ukraine.

Graduated: Moscow State Univ., Math. Dept., 1986.

Ph.D.: Moscow State Univ., 1992.

Over 20 research papers on Control Theory.

Professional Activity: Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalessky, 1989-present;

University of Pereslavl, 1993-present.

Steklov Inst. for Mathematics, Moscow, 1998-1999;

International School for Advanced Studies (SISSA-ISAS), Trieste, 1999-2001.

Current positions: Senior researcher of Program Systems Institute, Pereslavl-Zalessky, Russia;

Associate professor of University of Pereslavl, Russia.